Quaternionic modular forms and exceptional sets of hypergeometric functions

2015 ◽  
Vol 11 (02) ◽  
pp. 631-643 ◽  
Author(s):  
Srinath Baba ◽  
Håkan Granath
Keyword(s):  
Modular Forms ◽  
Hypergeometric Functions ◽  
Number Fields ◽  
Triangle Group ◽  
Exceptional Sets ◽  
Cm Points ◽  
Modular Polynomials

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

scholarly journals ON THE SPECIAL VALUES OF L-FUNCTIONS OF CM-BASE CHANGE FOR HILBERT MODULAR FORMS

2013 ◽  
Vol 56 (1) ◽  
pp. 57-63
Author(s):  
CRISTIAN VIRDOL
Keyword(s):  
Finite Order ◽  
Modular Forms ◽  
Number Fields ◽  
Base Change ◽  
Hilbert Modular Forms ◽  
Special Values ◽  
Finite Dimensional ◽  
Arbitrary Base ◽  
Hilbert Modular

AbstractIn this paper we generalize some results, obtained by Shimura, on the special values of L-functions of l-adic representations attached to quadratic CM-base change of Hilbert modular forms twisted by finite order characters. The generalization is to the case of the special values of L-functions of arbitrary base change to CM-number fields of l-adic representations attached to Hilbert modular forms twisted by some finite-dimensional representations.

scholarly journals On two arithmetic theta lifts

2018 ◽  
Vol 154 (10) ◽  
pp. 2090-2149 ◽  
Author(s):  
Stephan Ehlen ◽  
Siddarth Sankaran
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Chow Group ◽  
Shimura Varieties ◽  
Green Functions ◽  
Theta Lift ◽  
Moderate Growth ◽  
Generating Series ◽  
Cm Points ◽  
The Difference

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

ALGEBRAIC INDEPENDENCE OF VALUES OF MODULAR FORMS

2011 ◽  
Vol 07 (04) ◽  
pp. 1065-1074 ◽  
Author(s):  
SANOLI GUN ◽  
M. RAM MURTY ◽  
PURUSOTTAM RATH
Keyword(s):  
Recent Work ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Independence ◽  
Singular Values ◽  
Critical Values ◽  
Cm Points

We investigate values of modular forms with algebraic Fourier coefficients at algebraic arguments. As a consequence, we conclude about the nature of zeros of such modular forms. In particular, the singular values of modular forms (that is, values at CM points) are related to the recent work of Nesterenko. As an application, we deduce the transcendence of critical values of certain Hecke L-series. We also discuss how these investigations generalize to the case of quasi-modular forms with algebraic Fourier coefficients.

A NONVANISHING LEMMA FOR CERTAIN PADÉ APPROXIMATIONS OF THE SECOND KIND

2011 ◽  
Vol 07 (08) ◽  
pp. 1977-1997
Author(s):  
VILLE MERILÄ
Keyword(s):  
Diophantine Approximation ◽  
Algebraic Number ◽  
Hypergeometric Functions ◽  
Linear Independence ◽  
Number Fields ◽  
Vandermonde Determinant ◽  
Algebraic Number Fields ◽  
Padé Approximations ◽  
Pade Approximations

We prove the nonvanishing lemma for explicit second kind Padé approximations to generalized hypergeometric and q-hypergeometric functions. The proof is based on an evaluation of a generalized Vandermonde determinant. Also, some immediate applications to the Diophantine approximation is given in the form of sharp linear independence measures for hypergeometric E- and G-functions in algebraic number fields with different valuations.

scholarly journals Computing Modular Polynomials

2005 ◽  
Vol 8 ◽  
pp. 195-204 ◽  
Author(s):  
Denis Charles ◽  
Kristin Lauter
Keyword(s):  
Number Theory ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Probabilistic Algorithm ◽  
Computational Number Theory ◽  
Modular Polynomials

AbstractThis paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.

scholarly journals Nonsolvable number fields ramified only at 3 and 5

2011 ◽  
Vol 147 (3) ◽  
pp. 716-734 ◽  
Author(s):  
Lassina Dembélé ◽  
Matthew Greenberg ◽  
John Voight
Keyword(s):  
Modular Forms ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

AbstractFor p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

scholarly journals Eisenstein series in the Kohnen plus space for Hilbert modular forms

2016 ◽  
Vol 12 (03) ◽  
pp. 691-723 ◽  
Author(s):  
Ren-He Su
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
The One ◽  
Totally Real

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

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